Molecular polariton electroabsorption

We investigate electroabsorption (EA) in organic semiconductor microcavities to understand whether strong light-matter coupling non-trivially alters their nonlinear optical [\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\chi }^{(3)}\left(\omega,{{{{\mathrm{0,0}}}}}\right)$$\end{document}χ(3)ω,0, 0] response. Focusing on strongly-absorbing squaraine (SQ) molecules dispersed in a wide-gap host matrix, we find that classical transfer matrix modeling accurately captures the EA response of low concentration SQ microcavities with a vacuum Rabi splitting of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hslash \Omega \approx 200$$\end{document}ℏΩ≈200 meV, but fails for high concentration cavities with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hslash \Omega \approx 420$$\end{document}ℏΩ≈420 meV. Rather than new physics in the ultrastrong coupling regime, however, we attribute the discrepancy at high SQ concentration to a nearly dark H-aggregate state below the SQ exciton transition, which goes undetected in the optical constant dispersion on which the transfer matrix model is based, but nonetheless interacts with and enhances the EA response of the lower polariton mode. These results indicate that strong coupling can be used to manipulate EA (and presumably other optical nonlinearities) from organic microcavities by controlling the energy of polariton modes relative to other states in the system, but it does not alter the intrinsic optical nonlinearity of the organic semiconductor inside the cavity.


S2. Transfer matrix method for simulating EA spectra
The complex refractive index dispersion ( ) of the SQ films is described as a function of energy ( ) using a generic Lorentz oscillator model: where r = 1 + is the linear dielectric function, ∞ is the high frequency dielectric constant, and , , and Γ are the energy, amplitude, and broadening of the j-th oscillator, respectively. In the presence of an applied electric field, , the refractive index may be modified, most generally through a change in oscillator energy ( → + Δ ), amplitude ( → + Δ ), or broadening (Γ → Γ + ΔΓ ). From the perspective of nonlinear optics, the change in susceptibility that results, ∆ , is related to the third order nonlinear susceptibility via (&) ( , 0,0) = ∆ (3 ! ) ⁄ , where = ℏ ⁄ is the angular frequency 1 .
Using the transfer matrix model to calculate the sample reflectivity with and without the field-modified refractive index, it is then straightforward to obtain the differential reflectivity, ∆ ⁄ ≡ [ ( ) − (0)] (0) ⁄ that is measured in experiment. Other field-induced changes such as electrostriction (i.e. a field-induced decrease in cavity length) are also straightforward to account for, though we found no evidence that this is significant in our data analysis.
Additionally, the applied field could produce uniaxial anisotropy in the refractive index, but this would not impact the s-polarized reflectivity that we study here. Figure S2 shows the polariton dispersion relations obtained from the reflectivity data of the negatively-and positively-detuned 5 wt% SQ:NPB cavities in Fig. 2d,g, respectively. The data are fit using Eqn. (1) considering a single exciton state, which reduces to the standard 2x2

S5. Electronic structure calculations for squaraine
The impact of an applied field on the SQ exciton transition is examined by computing 'field-off' and 'field-on' linear response time-dependent density functional theory (TDDFT) excitation energies and transition dipole moments within the Tamm-Dancoff approximation. Geometry optimizations and normal mode analysis, dipole moment computation, as well as molecular orbital visualization and TDDFT calculations, are performed in vacuum using the Gaussian electronic structure package version 16 at the B3LYP/6-31G* level of theory. The / → * bright exciton transition is primarily HOMO to LUMO in character, with the molecular orbitals shown in Fig. S4. Note that the excitation energy is overestimated with TDDFT, partially due to the use of a relatively small basis set. With the molecular axis along the x-direction, the xcomponent of the transition dipole is the largest, and a field applied in this direction couples to the exciton transition. An applied field along the x-direction of +/-0.01 atomic units (a.u., equivalent to ~51 MV cm -1 ) shifts the energy of the transition by 0.03-0.04 eV; see Table S1.
Fields applied in the y-or z-direction cause minimal change to the transition energy. Figure S4. B3LYP/6-31G* molecular orbitals for computed for squaraine in vacuum. The S0 to S1 exciton transition is primarily HOMO to LUMO in character. operates with the static electric field and cavity coupling all at once. Though intuitively expected, it is not immediately obvious that these two different approaches yield equivalent results.
To show this, we begin by emphasizing the equivalence of the field-perturbed polariton modes computed in each basis. That is, the polariton energies obtained by diagonalizing Eqn. (1) in the light-matter basis (call them 67* , 67! , and 67& ) are equivalent to what one obtains by first diagonalizing the matter Hamiltonian: to determine the field-perturbed exciton eigenenergies, *,! , and then coupling these exciton states to the cavity photon via: :; where * < and ! < are the light-matter couplings projected in the field-perturbed matter basis (i.e. To calculate the electroreflectance spectra from the transfer matrix perspective, we determine the effect of the applied field on the two Lorentz oscillators by diagonalizing Eqn. (S3) with a field perturbation *! = 1 meV to determine their new energies ( *,! ) and amplitudes (i.e. *,! < = *,! W *,! < *,! ⁄ X ! since oscillator strength is proportional to the square of the light-matter interaction energy). Re-simulating the reflectivity with these new oscillator parameters then enables calculation of the differential reflectance spectra (relative to the zero-field spectra in Fig. S5b) shown in Fig. S5e. To calculate the electroreflectance from the perspective of Eqn. (1), we diagonalize it with and without the field perturbation to determine the associated polariton energy shifts plotted in Fig. S5d. We then fit the original zero-field reflectance in Fig. S5b with three Lorentzian peaks that capture the contribution of each polariton, apply their respective field-induced energy shifts from Fig. S5d, and then calculate the resulting differential reflectivity spectra, which exhibit good agreement with the transfer matrix results in Fig. 5e (blue dotted and black solid lines, respectively). Thus, we conclude that the two approaches to describe polariton EA are equivalent within the error of the multipeak fitting. We perform a similar numerical experiment to check the self-consistency of the reflectivity derivative approach used to determine the polariton energy shifts in Fig. 5 of the main text. For this example, we model the same scenario as in Fig. 5c,d, which involves field-induced mixing between the exciton at * = 1.85 eV and an upper excited state at ! = 3 eV. Figure S6a shows the optical constant dispersion in the vicinity of the exciton transition and Fig. S6b shows the angle-dependent reflectivity spectra that are simulated for a microcavity with 160 nm of this organic material placed between a 100 nm-thick bottom Ag mirror and a 30 nm-thick semitransparent top Ag mirror. Figure S6c shows the transfer matrix-simulated differential reflectance spectra following the procedure described above (i.e. based on the oscillator energy shift and amplitude change calculated from Eqn. (S3)), which are similar in lineshape but different in amplitude from the zero-field reflectivity derivative spectra shown in Fig. S6d. After aligning the Hamiltonian-simulated polariton dispersion with that obtained from the transfer matrix reflectivity data (Fig. S6e), we calculate the field-induced polariton shifts from Eqn. (1) and compare them with those extracted by scaling the peaks of the LP and UP reflectivity derivatives in Fig. S6d to match the mock EA data in Fig. S6c. The agreement between the two calculations shows that the reflectivity derivative method of inferring polariton energy shifts from EA spectra is self-consistent with the shifts obtained directly from the system Hamiltonian.  Figure S7a shows the normalized absorption spectra for a series of 100 nm-thick SQ:NPB films, highlighting an increase in the 0-1 high energy vibronic shoulder relative to the 0-0 transition with increasing SQ concentration . This is accompanied by a relative increase in the 0-1 vibronic photoluminescence intensity (Fig. S7b) as well as a rapid decrease in photoluminescence quantum yield (Fig. S7c). All of these observations are consistent with the signatures of Haggregate formation described in Ref. [2] and have also been observed for SQ in other host matrices 3 . We note that the broadening on the low energy side of the main exciton transition in

S8. Model including both H-aggregate states
The two-level model in the main text (Eqn. (1)) is readily expanded to include another (the upper H-aggregate) state at & = 1.94 eV:  Table S2 below.

S9. Effect of lower H-aggregate energy and oscillator strength
The bump in the LP shift at ~50º in Fig. S8d,f is associated with the point at which the LP mode crosses the lower H-aggregate state. Figure

S10. Impact of strong coupling on the field dependence of EA: SubPc cavities
Another interesting feature of Eqn. (1) is that it predicts a change in the field dependence of the EA signal when both transitions are strongly coupled to the cavity mode. In the absence of a cavity, Eqn. (1) reduces to: and has eigenvalues of: assuming that the field perturbation is much smaller than the zero-field energy difference between the two states (i.e. *! ≪ ! − * , assuming ! > * for concreteness). The fieldinduced energy shift of each state is thus ±( *! ) ! ( ! − * ) ⁄ , which scales quadratically with the field strength as expected from standard perturbation theory. A different approach to obtain the same result follows from expressing the eigenvalues as the sum of a zero-field component ( IJ , which in this case is just * and ! ) and the field-induced perturbation ( J ) via → IJ + J . Writing the characteristic eigenvalue equation and then setting the terms with lowest order (i.e. linear) in J equal to those involving the field, we obtain: which is equivalent to Eqn. (S7) above since IJ is just * or ! .
When both transitions are strongly coupled to the same cavity mode per Eqn. (1) from the main text, the same perturbative approach can be used to obtain the field-induced shift of the polariton modes: where now IJ are the energies of the zero-field lower, middle, and upper polaritons. It is evident from Eqn. (S9) that when only one of the two transitions strongly couples to the cavity mode, the polariton energy shifts scale quadratically with field, but when both of them couple, the shifts become linear in the field since * , ! ≫ | *! | (e.g. * , !~1 00 meV vs. *!~1 00 µeV). Physically, this nonzero ( ) can be understood by applying the static field perturbation to the zeroth-order polariton states; since the latter carry amplitudes from excitons 1 and 2, the static field mixes them already at first-order perturbation theory.
To explore this prediction, we study the EA response of cavities containing SubPc diluted at 25 wt% into the wide-gap host material CBP. SubPc is unique in this context because it has a doubly-degenerate * exciton transition 6 that should, in principle, fulfill the requirements for linear field scaling in Eqn. (S9). Its non-planar conformation also discourages aggregation, thereby minimizing the possibility of unforeseen aggregate state contributions to EA as in the case of SQ.
The impact of an applied field on the degenerate SubPc exciton transition is examined using the same TDDFT methodology as for SQ above. In vacuum, SubPc has C3 symmetry and degenerate lowest energy * and * < transitions. The dominant molecular orbital contributions to each transition are shown in Fig. S10. An applied electric field breaks the degeneracy of these excited states, with a field of 0.01 atomic units (a.u., equivalent to ~51 MV cm -1 ) leading to state splitting of 0.03-0.2 eV, depending on the direction of the field (see Table S3). Figure S10. B3LYP/6-311+G*(2d,p) molecular orbitals for SubPc in vacuum. The S1 transition is primarily HOMO to LUMO (left) and the S1' transition is primarily HOMO to LUMO (right).
The transition dipole moment strength between * and * < , computed with the configuration interaction singles CIS/6-311+G(2d,p) method, is K = 0.60 a.u. = 1.52 Debye, L = 0.25 a.u. = 0.65 Debye, and M = 0, indicating that a field applied along the x or y directions will couple the degenerate states, causing them to mix, with one state shifting up in energy and the other shifting down in energy. Both the ground and excited state dipole moments are primarily along the z-direction (the direction of the C3 axis), with the excited state dipole moment increasing in the y and zdirections, leading to preferential stabilization of the excited state over the ground state if the field is aligned with the dipole moment.
The experimental data in Fig. S11 support these predictions. Figure S11a shows the extinction coefficient spectrum measured for a 25 wt% SubPc:CBP film, which exhibits a primary 0-0 exciton transition at ~2.12 eV and a vibronic shoulder at ~2.3 eV that technically involves several vibrations 6 , but which we refer to as 0-1 for simplicity. As with the SQ devices in the main text, we initially characterize the angle-dependent reflectivity and electroreflectance of a half-cavity control device (120 nm ITO/5 nm MoO3/130 nm 25 wt% SubPc:CBP/20 nm Ag) shown in Fig. S11b and S11c, respectively. Consistent with the TDDFT results above, we find good agreement with the transfer-matrix-simulated EA in Fig. S11c when we assume that the SubPc absorption derives from two, nearly degenerate electronic transitions, * and * < , that are each modeled by a pair of Lorentz oscillators (denoted by the purple and green dashed lines in . Although a change in broadening might also be expected given the change in static dipole moment predicted by the TDDFT calculations, it is not required to reproduce the data and thus must be a small effect compared to the energy and amplitude changes. We note as well that, although the excited state degeneracy is exact for an isolated SubPc molecule in Table S3 above, we have to assume a 30 meV zero-field splitting between the * and * < oscillators in Fig. S11a in order to reproduce the EA data. This zero-field splitting presumably stems from molecular packing effects in the film that distort the SubPc nuclear framework and break the formal excited state degeneracy of the isolated molecule.   The green markers indicated by asterisks correspond to the reflectivity shoulder features identified in Fig. S12a. (b) Field-induced LP shift derived from the reflectivity derivative analysis in Fig. S12c Figure S13b shows that the field-induced LP shift is reasonably well captured by the model; comparison of the UP shift is unfortunately hindered by the difficulties with its reflectivity derivative lineshape noted above. For this reason, we focus our investigation on the field dependence of polariton EA exclusively on the LP mode.
To understand whether the field dependence of the LP EA differs from that of the halfcavity control, we monitor the ratio of their first-to-second harmonic EA amplitude as a function of DC bias. For a generalized EA response potentially having both linear and quadratic contributions to the field dependence, ∆ = + ! + ⋯, and assuming an applied electric field = Q) + R) sin( ), it is straightforward to show that the ratio of the first-to-second harmonic magnitudes, ∆ *S ∆ !S ⁄ = 2 ( R) ) ⁄ + 4 Q) R) ⁄ . Thus, plotting ∆ *S ∆ !S ⁄ versus Q) for different fixed values of R) should produce a family of lines that all intercept the x-axis at Q) = − (2 ) ⁄ , thereby providing a measure of the relative contributions of the linear and quadratic components of the EA response.  amplitudes (implying that = 0 and the field dependence is fully quadratic), the intercepts for the LP feature in the cavity device seem to cluster away from the origin, implying a non-zero linear contribution to the signal. There is, however, significant scatter in the LP intercepts and thus it is not possible for us to conclude with certainty whether the LP EA truly exhibits a different field dependence than the control, though it does seem clear that the LP field dependence has a significant quadratic contribution since otherwise there would be no second harmonic signal at all. A full understanding of the field dependence of polariton EA in relation to that predicted by Eqn. (S9) thus remains a key question for future work.